WEBVTT 00:18.093 --> 00:28.863 When the leading coefficient of a quadratic expression is something other than one, 00:28.920 --> 00:35.169 there are two methods that we can use to factor the quadratic expression. 00:35.220 --> 00:40.975 The first method that I'm going to show you is called trial and error. 00:41.040 --> 00:52.620 Let's look at our first problem. Three r squared plus sixteen r plus five. 00:52.680 --> 01:02.897 This method of trial and error is the most efficient method when your values of a and c 01:02.940 --> 01:17.111 don't have many factors. So we're first going to begin by listing factors of five. Factors of five 01:17.160 --> 01:31.025 are plus and minus one and plus and minus five. Next, we will look at factors of a, or factors of three. 01:31.058 --> 01:44.038 I know that factors of three are plus and minus one and plus and minus three. 01:44.100 --> 01:48.976 Since we don't have a long list of factors for each of these numbers, 01:49.009 --> 02:03.858 trial and error is the method that will be most efficient. With trial and error, we are going to 02:03.900 --> 02:14.835 make our linear factors, and what we're looking for here is knowing that when we multiply these 02:14.869 --> 02:24.612 two numbers together we need a positive five. And so because we need a positive five, 02:24.660 --> 02:37.992 these numbers could either be negative and negative, or they could be positive and positive. 02:38.040 --> 02:51.805 But now we need to look more closely at our value of b. Since our value of b is a positive sixteen, the only 02:51.839 --> 03:13.794 way for us to get a positive sixteen sum and a product of positive five is if both of these numbers are positive. 03:13.860 --> 03:20.401 Now I know the only way for us to multiply these two together and 03:20.460 --> 03:36.016 get a positive five is if one of those numbers is five and the other is one. Now we need to figure out what these need to be 03:36.060 --> 03:49.396 so that when we multiply them together we will get a positive three r squared. 03:49.440 --> 04:03.310 Since it's r squared, I know that I will have to have an r in each of my factors, and the only way for us to get three r squared 04:03.360 --> 04:13.587 is if one of these numbers is three and the other is just one. So let's check to see if this is 04:13.620 --> 04:26.066 the correct factored form for this trinomial. Three r times r does give me three r squared. 04:26.100 --> 04:39.179 But notice three r plus five r does not add to give me sixteen r. Therefore, I need to know 04:39.240 --> 04:51.959 or I notice that this is not the correct factored form. So let's try another one. 04:52.020 --> 05:03.303 Again, I know that we will have to have a positive five and a positive one. 05:03.360 --> 05:12.980 But since three in this term and one in that term did not work, let's 05:13.013 --> 05:28.495 try one r there and three r here. To check, r times three r, three r squared. 05:28.560 --> 05:46.146 This is r plus fifteen r which does give me my sixteen r, and positive five times positive one does give me positive five. 05:46.200 --> 06:00.661 Therefore, this is the correct factored form of this trinomial. Now let's take a look at our second one. 06:00.720 --> 06:18.445 Again, notice your value of a is some number other than one. Again, I'm looking at factors of c as well as factors of a. 06:18.480 --> 06:27.621 For my factors of negative one, I know that that's just plus or minus one. 06:27.660 --> 06:39.133 And in terms of factors of four, I know that those are plus and minus one 06:39.180 --> 06:51.245 and plus and minus four as well as plus or minus two. So let's start trying to see which 06:51.300 --> 07:10.197 way we can factor this trinomial. Now since when multiplying these two numbers together I need to get a negative one, 07:10.260 --> 07:15.869 that tells me the only way to multiply two numbers together to get 07:15.903 --> 07:26.713 negative one is if one of them is positive one and one of them is negative one. 07:26.760 --> 07:39.126 Now let's try to work with these. I need four x squared, positive four x squared. One way 07:39.180 --> 07:46.166 to get positive four x squared is a positive two x and a positive two x. 07:46.200 --> 07:55.576 Now let's check to see if that is the correct form. Four x squared. Minus two x plus two x, 07:55.620 --> 08:08.121 that makes zero, not positive three. Therefore, this is not the correct factored form. 08:08.160 --> 08:19.700 Now we can try another set of factors of four. Again, this will have to be positive one and 08:19.740 --> 08:35.115 negative one. But now we can try a positive four x and a positive one x. Let's see if that's it. Four x squared. 08:35.160 --> 08:51.598 Minus four x plus x gives me a negative three x. That's not it. I need positive three x. 08:51.660 --> 09:04.011 So the only other possible way this could factor, again keeping the plus one and 09:04.044 --> 09:16.390 the minus one, this time let's try the one x here and the four x here. 09:16.423 --> 09:30.370 Four x squared. Minus x plus four x, there's my positive three x. And one times negative one is 09:30.420 --> 09:47.120 negative one. That means this, and not the previous two, is factored form of this trinomial. 09:47.160 --> 09:54.428 The second method that we can use in factoring a trinomial with a coefficient other than one 09:54.480 --> 10:04.404 is sometimes referred to as splitting the linear term. Let's take a look at our first example. Seven 10:04.440 --> 10:17.584 m squared minus nineteen m minus six. To begin this process, we are going to multiply a times c. 10:17.640 --> 10:29.362 Seven times negative six gives me negative forty-two. So a times c is equal to negative forty-two. 10:29.400 --> 10:44.845 It's also important to know the value of b which in this case is negative nineteen. Now we're looking for a pair of numbers 10:44.880 --> 10:55.088 whose product is negative forty-two and whose sum is negative nineteen. Let's begin by listing 10:55.140 --> 11:05.899 the factors of negative forty-two. Those are plus or minus one, plus or minus forty-two, 11:05.932 --> 11:15.375 plus and minus two, plus and minus twenty-one, plus and minus three, 11:15.408 --> 11:25.819 plus and minus fourteen, and our last set of factors plus or minus six and plus or minus seven. 11:25.860 --> 11:33.493 Now of this list of factors, we're most interested in the pair that adds to give 11:33.540 --> 11:47.574 us negative nineteen. Hopefully you can recognize that that is positive two and negative twenty-one. 11:47.640 --> 11:55.081 Now what we're going to do is we're going to split our linear term using 11:55.140 --> 12:13.567 these two numbers, so instead of negative nineteen m, I'm going to split my linear term into two m minus twenty-one m. 12:13.620 --> 12:21.074 Please recognize that rewriting it this way is not changing the value of the expression. 12:21.120 --> 12:34.821 It's simply rewriting it in a different form. I'm going to keep c, and I'm going to keep a. 12:34.860 --> 12:43.697 Now notice we have four terms, and we can factor four terms using the grouping method. 12:43.740 --> 12:51.504 I recognize that my first two terms do have a common factor, and I also recognize that 12:51.540 --> 12:59.179 my second two terms have a common factor. So I'm going to include 12:59.220 --> 13:08.588 the negative with the twenty- one, and I'm going to put addition between them. The common 13:08.640 --> 13:22.135 factor of my first group is m. Factoring out an m will leave me with seven m plus two. 13:22.200 --> 13:30.844 In my second group, I notice that there is a common factor of three. However, 13:30.900 --> 13:39.252 if I'm factoring out a positive three, then that will give me different signs than 13:39.300 --> 13:49.696 a positive seven m and a positive two. I'm going to factor out a negative three. 13:49.740 --> 14:00.740 Factoring a negative three out of this group will give me a positive seven m plus two. Now if you recall 14:00.774 --> 14:11.818 from the grouping method, both of these terms have a common factor of seven m plus two, 14:11.880 --> 14:23.596 and once I factor out seven m plus two from each of the terms, I am left with m minus three. 14:23.640 --> 14:33.673 And this is factored form of this trinomial. Let's look at our second example. Eight 14:33.720 --> 14:51.858 w squared minus eighteen w plus nine. When I multiply a times c, I get seventy-two. 14:51.900 --> 15:05.338 I notice that b is negative eighteen. Again, we're looking for factors of seventy-two 15:05.400 --> 15:16.149 whose sum is negative eighteen. So let's list the factors of seventy-two. They are plus or minus one 15:16.182 --> 15:26.092 and plus or minus seventy-two, plus or minus two and plus or minus thirty-six, 15:26.126 --> 15:41.007 plus or minus three and plus or minus twenty-four, plus or minus four, plus or minus eighteen, 15:41.040 --> 15:49.682 plus or minus six, plus or minus twelve, and finally plus and minus eight and plus 15:49.740 --> 16:01.394 and minus nine. Of these pairs of factors, the pair whose sum is also negative eighteen 16:01.440 --> 16:16.142 is negative twelve and negative six. So now I'm going to split my linear term into 16:16.200 --> 16:31.224 negative twelve w minus six w. I'm going to keep my first term, and I will keep my last. 16:31.260 --> 16:38.665 Again, recognizing that we now have four terms, I can now factor by grouping. 16:38.700 --> 16:50.109 I see that my first two terms do share a common factor as do my second two. 16:50.160 --> 17:06.793 The greatest common factor of the first group is four w, and when I factor out four w, I'm left with two w minus three. 17:06.840 --> 17:19.739 The greatest common factor of my second group happens to be three. However, if I factor out a positive three, 17:19.800 --> 17:28.915 then that will leave me with a negative two w and a positive three. I need a positive two w and a 17:28.980 --> 17:39.659 negative three. So I'm going to factor out a negative three. And when I factor out a negative three, I do indeed end up 17:39.692 --> 17:51.237 with two w minus three. Recognizing that both of these terms share a common factor of two w 17:51.300 --> 18:02.815 minus three, I can then factor out the two w minus three, and that makes my other linear factor 18:02.880 --> 18:19.899 four w minus three. And this is factored form of this trinomial. We're going to continue our practice of 18:19.932 --> 18:33.913 splitting the linear term with this next example. Let's factor twelve x cubed plus ten x squared plus two x. 18:33.960 --> 18:45.124 Notice that we do have a value of a that's other than one, but I also want you to notice 18:45.180 --> 18:53.040 that looking at all three of your terms you have a greatest common factor, and we should always 18:53.066 --> 19:02.775 begin the factoring process looking to see if we can factor out a greatest common factor. 19:02.820 --> 19:14.420 Looking at all three terms, I recognize that my greatest common factor is two x. 19:14.460 --> 19:20.293 If I factor out a two x from each of the terms, I'm going to be left with 19:20.326 --> 19:31.704 six x squared plus five x plus one. Now we're going to continue to factor this 19:31.740 --> 19:39.445 trinomial, and we're going to focus on the trinomial inside of the parentheses. 19:39.480 --> 19:46.085 This trinomial also has a leading coefficient other than one. So now 19:46.119 --> 19:52.458 we can choose whether to use the trial and error method or splitting the 19:52.500 --> 20:06.606 linear term. With a number like six which has more factors than say two, three, or four, or five, 20:06.660 --> 20:14.480 trial and error may not be the most efficient method in factoring this trinomial. 20:14.520 --> 20:20.653 So let's go ahead and use the method of splitting the linear term. 20:20.700 --> 20:41.174 a times c is six, and b is five. So I'm looking for factors of six who also add to give me five. 20:41.220 --> 20:57.056 Factors of six: plus or minus one, plus or minus six, plus or minus two, and plus or minus three. 20:57.120 --> 21:12.138 Of these factors, the pair which also adds to give me five is positive two and positive three. 21:12.180 --> 21:26.419 Therefore, I am now going to split my linear term and rewrite it as two x plus three x. I'm going to 21:26.460 --> 21:39.765 keep the one. I'm also going to keep the six x squared. Now I know that I have this factor, this greatest common factor 21:39.799 --> 21:51.010 of two x here which in the end we will absolutely put as part of completely factored form. Right now 21:51.060 --> 22:00.686 though, we're going to continue factoring these four terms using the grouping method. 22:00.720 --> 22:08.895 My first two terms have a common factor, and some of you may think the last two don't. 22:08.940 --> 22:15.501 However, they do, and we'll get to that in just a second. The common factor 22:15.534 --> 22:25.278 between six x squared and two x is two x, and if I factor out a two x, I'm left 22:25.320 --> 22:36.622 with three x plus one. Now let's talk about the common factor between three x and one. 22:36.660 --> 22:47.700 That's one. And so factoring out a one from three x plus one will give me three x plus one. 22:47.760 --> 22:58.477 Again, I notice a common factor here of three x plus one, and when factoring that from each of the terms, 22:58.511 --> 23:10.823 I'm left with two x plus one. Let's not forget our original greatest common factor of two x. 23:10.860 --> 23:20.032 And this is factored form of this trinomial.